The modification of connections between neurons, known as synaptic plasticity, is crucial for experience-dependent development and tuning of neuronal circuits during the critical period after birth or in adult life. Spike-timing-dependent plasticity (STDP) has been uncovered more than a decade ago and describes how the precise timing of pre- and postsynaptic spikes modifies the synaptic strength. A number of computational functions have been suggested for STDP, e.g., detection of millisecond-precise spike sequences, independent component analysis, principal component analysis, formation of cell assemblies and self-organization of ongoing activity. However, all of them rely to some degree on competition between synapses, which is not fully compatible with the experimentally observed 'weight dependence' of STDP. By this, we mean how the STDP update depends on the current strength of the synaptic weight. Here, we propose a new model of weight-dependent STDP that is inferred from the recent experimental finding of lognormal-like distributions of synaptic weights. This new rule exhibits a remarkable flexibility in learning dynamics. Our results provide a firm theoretical ground for activity-dependent development of neuronal circuits. This work also stresses the importance of weight dependence for more elaborate STDP models (e.g., voltage-dependent or based on calcium kinetics), which has not been specifically addressed until now to our knowledge.